The existence of a symmetric global minimizer on [formula omitted] implies the existence of a counter-example to a conjecture of De Giorgi in [formula omitted

Abstract

For solutions of semilinear elliptic equations, we show the link that exists between global minimizers on R n−1 and the existence of nontrivial monotonic solutions on R n . We prove that the existence of a symmetric global minimizer on R n−1 implies the existence of a nonplanar monotonic solution on R n which is a global minimizer. Moreover, we prove that every monotonic solution on R n is a global minimizer if and only if its up and down limits are global minimizers.